as the best error bound with respect to the \(\Vert  \cdot \Vert _Q\) -norm for approximating a point \(\mathbf {z}\) 
in the convex hull of some points in \(P\) . We would like to point out that the vectors \(\mathbf {v}_1,\ldots ,\mathbf {v}_k\)  may be taken with repetition.
A folklore result is that for the Euclidean norm one has \(\textrm {ac}_k(B_2^m,B_2^m) \le \frac{1}{\sqrt{k}}\)  for any \(k \ge 1\) , which gives a dimension free bound. More generally, for \(p \ge 1\) , it is true that \(\textrm {ac}_k(B_p^m,B_p^m) \le O(\sqrt{\frac{p}{k}})\)  where \(B_p^m := \lbrace  \mathbf {x} \in \mathbb {R}^m \mid \Vert \mathbf {x}\Vert _p \le 1\rbrace \)  is the \(\Vert  \cdot \Vert _p\) -unit ball.
This bound is derived from
Maurey's Lemma from functional analysis (which was reported by Pisier [1]}; for an English version, see the appendix of Bourgain and Nelson [2]}).
Algorithmically, the result is simple: write \(\mathbf {z} = \sum _{i=1}^{N} \lambda _i\mathbf {u}_i\)  where \(\lambda _1,\ldots ,\lambda _N \ge 0\) 
and \(\sum _{i=1}^N \lambda _i=1\) . Then sample \(\mathbf {v}_1,\ldots ,\mathbf {v}_k \in \lbrace  \mathbf {u}_1,\ldots ,\mathbf {u}_N\rbrace \)  independently according to the probabilities \(\lambda _i\)  (possibly with repetition).
